...it's the situation as I know it to be from my experience of writing historical biographies. This is the law of copyright in the UK - the law makes a clear distinction between published and unpublished material. Unpublished material remains in copyright, so if you want to quote from an unpublished letter or diary you need to have permission no matter how old it is. In practice, if it's extremely old, the permission will normally come from the institution where the letter is kept, though when the family still hold letters they still have the copyright - on something like the Verney papers, say, which go right back to the seventeenth century (though most of those have been published already).

In this case, with some of the letter writers still alive and a book covering events which are well within living memory there is no way they could publish without permission. They'd have no case at all in law and they know it.